Your problem is with the bank, not the IRS. If you are keeping the business "open" but are not engaged in ongoing business activities, so that you don't have income or expenses to declare, you would usually not be required to file a schedule C as part of your tax return. However, if you have property you were depreciating, that is still part of the business even though the business is not active, that needs to be dealt with one way or another. You can't just ignore business property you are in the middle of depreciating if the business takes a break.
There may be state requirements to keep an LLC active, like paying a yearly registration fee or filing forms, we can't tell you about individual state requirements.
If you are not engaged in ongoing business activities, and not filing schedule C, you need to talk to someone at the bank about why they are trying to close the business account and what they want you to do. Of course, if you are not engaged in business activities, you don't need a business account, do you? Could you close it and open another account when you need it (maybe at a different bank)? And, since a single member LLC is a disregarded entity, there is no tax reason you can't run the business from a personal account—although it does make it easier to keep track of finances and is always a good idea to have separate accounts, it is not a legal requirement.
And if you are running money through the business account but not filing a schedule C, that could raise some questions about what you are really doing with the account.
